Preheat oven to 350 degrees. Grease a 9-inch round cake pan. Dust with flour; tap out excess. In a small bowl, mix together the flours, cinnamon, baking powder, and salt. Whisk thoroughly.

In the bowl of a stand mixer fitted with the beater attachment, beat together sugar and butter on medium speed until light and fluffy. Add eggs, one at a time, making sure each one is combined before adding another. On low speed, add in flour mixture and milk into butter mixture and mix until combined. Stir in carrots and walnuts, if using. Pour batter into prepared pan.

Bake for 45 minutes, or until a toothpick inserted in the center of the cake comes out clean. Transfer pan to a wire rack to cool for 10 minutes. Turn cake out onto rack to cool completely.

Beat together the butter and cream cheese on medium speed until light and fluffy. Add in the powdered sugar and vanilla extract until combined.

Place cake on a serving plate or cake stand. Spread cream cheese frosting over the top and sides.